鲁棒主成分分析关键技术研究及应用

主成分分析（principal component analysis, PCA）通过求解均方误差最小得到数据的有效描述，已被广泛地应用到数据分析和模式识别等领域。由于均方误差过分强调分布比较远的数据，导致鲁棒性差。对此问题，低秩PCA 和L1范数PCA是两种代表性的技术。然而，低秩PCA不能很好地保持数据的内在几何结构；而L1范数的解与离散度无关，不能使重构误差最小，且忽略了低维描述与误差的关系，导致鲁棒性不好。随着信息采集技术的发展，实际获取的数据不可避免的含有Outliers，使得他们远离真实分布，导致上述问题越来越突出。因此，如何提高PCA的鲁棒性已成为目前急需要解决的主要研究课题之一。基于此，本项目主要研究自适应邻接图的鲁棒低秩PCA、角度PCA、核角度PCA以及张量角度PCA。项目研究内容是模式识别、机器学习等领域非常活跃的研究方向之一，具有非常重要的理论价值和应用价值.

Principal component analysis (PCA) can well obtain effective representation by least mean squared errors and has been widely used in the fields of data analysis and pattern recognition. Since least mean squared error remarkably enlarge the large distance, PCA is not robust to noise. To improve robustness of PCA, many methods have been developed, among which low-rank PCA and L1-norm PCA are two of the most representative techniques. However, low-rank PCA does not well preserve the local intrinsic structure, which is important for data representation. L1-norm PCA does not relate to the covariance matrix of data and ignores the relationship between low-dimensional representation and reconstruction error. Moreover, L1-norm PCA does not reveal the nonlinear features. These affect the robustness of PCA. With the development of information technology, data inevitably contains noise, this makes the aforementioned problem more and more prominent in real applications. Thus, how to improve the robustness of PCA to outliers becomes one of the very important problems that urgently needs to be solved. Based on it, we study the adaptive adjacency graph based robust low-rank PCA, angle PCA and robust kernel based angle PCA. These contents of this project are very important in pattern recognition and machine learning, and will help to promote the development of the related areas.